Abstract
In this paper, we investigate the interpolation properties of discrete net spaces np,q(M) and examine their applications to the analysis of linear operators acting on these spaces. These spaces are characterized by the property that, for monotonically non-increasing sequences, the norm in np,q(M) coincides with the norm of the discrete Lorentz space lp,q(M). At the same time, unlike Lorentz spaces, these spaces np,q(M) may contain sequences that do not tend to zero, making them suitable for the study of more general function spaces and operator classes. The main result of this paper is the proof of the interpolation theorem for discrete net spaces with respect to the real interpolation method, providing a systematic framework for estimating norms of intermediate spaces. It is shown that the scale of discrete net spaces np,q(M) is closed under the real interpolation method, ensuring the stability under interpolation procedures. Furthermore, we present an analogue of Marcinkiewicz’s interpolation theorem, which allows us to establish strong boundedness of linear operators in discrete net spaces based on their weak boundedness on local nets. These results contribute to a deeper understanding of the structure of discrete net spaces and offer new tools for applications in harmonic analysis, operator theory, and related areas where such spaces naturally arise.